Understanding Series Divergence

The question asks us to find the condition under which the given series is divergent. Understanding the behavior of infinite series, whether they converge to a finite sum or diverge, is a fundamental concept in calculus and mathematical analysis.

Identifying the Series Type: The p-series

The given series is \(\sum {\left( {\frac{1}{{np}}} \right)}\). While the notation might seem slightly ambiguous, in the context of standard series tests and the provided options involving the parameter \(p\), this series is almost certainly intended to be interpreted as the p-series:

\[ \sum_{n=1}^{\infty} \frac{1}{n^p} \]

A p-series is a specific type of infinite series that has the form \(\sum_{n=1}^{\infty} \frac{1}{n^p}\), where \(p\) is a positive real number. The convergence or series divergence of a p-series depends entirely on the value of \(p\).

The p-series Convergence Test

There is a well-established test, known as the p-series test or the hyperharmonic series test, which determines the convergence or series divergence of a p-series. This convergence test states:

• The series \(\sum_{n=1}^{\infty} \frac{1}{n^p}\) converges if \(p > 1\).
• The series \(\sum_{n=1}^{\infty} \frac{1}{n^p}\) diverges if \(p \le 1\).

This is a very useful convergence test for quickly analyzing series of this form.

Applying the Test to Find Series Divergence

We are interested in the condition for the series \(\sum \frac{1}{n^p}\) to be divergent. According to the p-series convergence test mentioned above, the series diverges when the exponent \(p\) is less than or equal to 1.

So, the condition for the series divergence of \(\sum \frac{1}{n^p}\) is \(p \le 1\).

Comparing with Options

Let's look at the given options:

• Option 1: \(p \ge 1\). This includes \(p=1\) where it diverges and \(p > 1\) where it converges. This is not the complete condition for divergence.
• Option 2: \(p < 1\). This is a part of the condition for divergence (\(p \le 1\)), but it excludes the case \(p=1\), where the series also diverges (the harmonic series \(\sum \frac{1}{n}\)).
• Option 3: \(p \le 1\). This matches the exact condition derived from the p-series convergence test for series divergence.
• Option 4: None of these. This is incorrect as Option 3 is correct.

Therefore, based on the p-series test, the series \(\sum \frac{1}{n^p}\) is divergent if and only if \(p \le 1\). This is the key condition for the series divergence of this type of mathematical series.

Conclusion on Series Divergence

The series \(\sum {\left( {\frac{1}{{np}}} \right)}\), interpreted as the p-series \(\sum \frac{1}{n^p}\), is divergent when the value of the exponent \(p\) is less than or equal to 1. This conclusion is directly derived from the p-series convergence test, a standard result in the study of infinite series.